Difference between revisions of "1981 AHSME Problems/Problem 16"
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==Solution (Long Way)== | ==Solution (Long Way)== | ||
Convert <math>x</math> to base 10 then convert the result to base 9. | Convert <math>x</math> to base 10 then convert the result to base 9. | ||
| − | <cmath>12112211122211112222_{3} = | + | <cmath>12112211122211112222_{3} = 2150029898</cmath> |
| − | <cmath> | + | <cmath>2150029898 = 5484584488_{9}</cmath> |
| − | Therefore, the answer is <math> \textbf{( | + | Therefore, the answer is <math> \textbf{(E)}\ 5.</math> |
-edited by coolmath34 | -edited by coolmath34 | ||
| + | |||
| + | ==Solution (Faster Way)== | ||
| + | Every 2 numbers in base 3 represents 1 number in base 9. | ||
| + | The first 2 numbers on the left,12 = 1(3) + 2(1) = 5. | ||
| + | |||
| + | So the answer is <math> \textbf{(E)}\ 5.</math> | ||
Latest revision as of 19:56, 5 May 2021
Problem
The base three representation of 
is
![\[12112211122211112222\]](http://latex.artofproblemsolving.com/1/9/e/19ee9a81d71f36177bfbd8f9886022b1ae264e14.png)
The first digit (on the left) of the base nine representation of is
Solution (Long Way)
Convert to base 10 then convert the result to base 9.
![\[12112211122211112222_{3} = 2150029898\]](http://latex.artofproblemsolving.com/6/4/e/64ebb09efcb51d16c956f98d227a8502de88a8b0.png)
![\[2150029898 = 5484584488_{9}\]](http://latex.artofproblemsolving.com/9/d/e/9ded47e3858d6a4a0d1be36ac3ce6e923e00ac6f.png)
Therefore, the answer is 
-edited by coolmath34
Solution (Faster Way)
Every 2 numbers in base 3 represents 1 number in base 9. The first 2 numbers on the left,12 = 1(3) + 2(1) = 5.
So the answer is
